Galactic winds driven by active galactic nuclei (AGN) have been invoked to play a fundamental role in the co-evolution between supermassive black holes and their host galaxies. Finding observational evidence of such feedback mechanisms is of crucial importance and it requires a multi-wavelength approach in order to compare winds at different scales and phases. In Tombesi et al. (2015) we reported the detection of a powerful ultra-fast outflow (UFO) in the Suzaku X-ray spectrum of the ultra-luminous infrared galaxy IRAS F11119$+$3257. The comparison with a galaxy-scale OH molecular outflow observed with Herschel in the same source supported the energy-conserving scenario for AGN feedback. The main objective of this work is to perform an independent check of the Suzaku results using the higher sensitivity and wider X-ray continuum coverage of NuSTAR. We clearly detect a highly ionized Fe K UFO in the 100ks NuSTAR spectrum with parameters $N_H = (3.2pm1.5)times 10^{24}$ cm$^{-2}$, log$xi$$=$$4.0^{+1.2}_{-0.3}$ erg s$^{-1}$ cm, and $v_{text{out}} = 0.253^{+0.061}_{-0.118}$ c. The launching radius is likely at a distance of $r ge 16$ $r_s$ from the black hole. The mass outflow rate is in the range $dot{M}_{out}$$simeq$0.5-2 $M_{odot}$ yr$^{-1}$. The UFO momentum rate and power are $dot{P}_{out} simeq$0.5-2 $L_{AGN}/c$ and $dot{E}_{out} simeq$7-27% $L_{AGN}$, respectively. The UFO parameters are consistent between the 2013 Suzaku and the 2015 NuSTAR observations. Only the column density is found to be variable, possibly suggesting a clumpy wind. The comparison with the energetics of molecular outflows estimated in infrared and millimeter wavelengths support a connection between the nuclear and galaxy-scale winds in luminous AGN.
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